### Chap004

```Chapter 4
Discounted Cash Flow Valuation
McGraw-Hill/Irwin
Key Concepts and Skills




Be able to compute the future value and/or
present value of a single cash flow or series of
cash flows
Be able to compute the return on an
investment
Be able to use a financial calculator and/or
spreadsheet to solve time value problems
Understand perpetuities and annuities
McGraw-Hill/Irwin
Chapter Outline
4.1 Valuation: The One-Period Case
4.2 The Multiperiod Case
4.3 Compounding Periods
4.4 Simplifications
4.5 What Is a Firm Worth?
McGraw-Hill/Irwin
4.1 The One-Period Case

If you were to invest \$10,000 at 5-percent interest
for one year, your investment would grow to
\$10,500.
\$500 would be interest (\$10,000 × .05)
\$10,000 is the principal repayment (\$10,000 × 1)
\$10,500 is the total due. It can be calculated as:
\$10,500 = \$10,000×(1.05)

The total amount due at the end of the investment is
call the Future Value (FV).
McGraw-Hill/Irwin
Future Value

In the one-period case, the formula for FV can
be written as:
FV = C0×(1 + r)T
Where C0 is cash flow today (time zero), and
r is the appropriate interest rate.
McGraw-Hill/Irwin
Present Value

If you were to be promised \$10,000 due in one year
when interest rates are 5-percent, your investment
would be worth \$9,523.81 in today’s dollars.
\$9,523.81 
\$10,000
1.05
The amount that a borrower would need to set aside
today to be able to meet the promised payment of
\$10,000 in one year is called the Present Value (PV).
Note that \$10,000 = \$9,523.81×(1.05).
McGraw-Hill/Irwin
Present Value

In the one-period case, the formula for PV can
be written as:
PV 
C1
1 r
Where C1 is cash flow at date 1, and
r is the appropriate interest rate.
McGraw-Hill/Irwin
Net Present Value


The Net Present Value (NPV) of an
investment is the present value of the
expected cash flows, less the cost of the
investment.
Suppose an investment that promises to pay
\$10,000 in one year is offered for sale for
\$9,500. Your interest rate is 5%. Should you
McGraw-Hill/Irwin
Net Present Value
NPV  \$9,500 
\$10,000
1.05
NPV  \$9,500  \$9,523.81
NPV  \$23.81
The present value of the cash inflow is greater
than the cost. In other words, the Net Present
Value is positive, so the investment should be
purchased.
McGraw-Hill/Irwin
Net Present Value
In the one-period case, the formula for NPV can be
written as:
NPV = –Cost + PV
If we had not undertaken the positive NPV project
considered on the last slide, and instead invested our
\$9,500 elsewhere at 5 percent, our FV would be less
than the \$10,000 the investment promised, and we
would be worse off in FV terms :
\$9,500×(1.05) = \$9,975 < \$10,000
McGraw-Hill/Irwin
4.2 The Multiperiod Case

The general formula for the future value of an
investment over many periods can be written
as:
FV = C0×(1 + r)T
Where
C0 is cash flow at date 0,
r is the appropriate interest rate, and
T is the number of periods over which the cash is
invested.
McGraw-Hill/Irwin
Future Value


Suppose a stock currently pays a dividend of
\$1.10, which is expected to grow at 40% per
year for the next five years.
What will the dividend be in five years?
FV = C0×(1 + r)T
\$5.92 = \$1.10×(1.40)5
McGraw-Hill/Irwin
Future Value and Compounding

Notice that the dividend in year five, \$5.92,
is considerably higher than the sum of the
original dividend plus five increases of 40percent on the original \$1.10 dividend:
\$5.92 > \$1.10 + 5×[\$1.10×.40] = \$3.30
This is due to compounding.
McGraw-Hill/Irwin
Future Value and Compounding
\$1.10  (1.40)
\$1.10  (1.40)
\$1.10  (1.40)
\$1.10  (1.40)
5
4
3
2
\$1.10  (1.40)
\$1.10
\$1.54
\$2.16
\$3.02
\$4.23
\$5.92
0
1
2
3
4
5
McGraw-Hill/Irwin
Present Value and Discounting

How much would an investor have to set
aside today in order to have \$20,000 five
years from now if the current rate is 15%?
PV
\$20,000
0
1
\$9,943.53 
2
4
5
\$20,000
(1.15)
McGraw-Hill/Irwin
3
5
How Long is the Wait?
If we deposit \$5,000 today in an account paying 10%,
how long does it take to grow to \$10,000?
FV  C0  (1  r )
\$10,000  \$5,000  (1.10)
T
(1.10) 
T
\$10,000
T
2
\$5,000
ln( 1.10)  ln( 2)
T
T
ln( 2)
ln( 1.10)
McGraw-Hill/Irwin

0.6931
 7.27 years
0.0953
What Rate Is Enough?
Assume the total cost of a college education will be
\$50,000 when your child enters college in 12 years.
You have \$5,000 to invest today. What rate of interest
must you earn on your investment to cover the cost of
FV  C0  (1  r )
(1  r )
12

T
\$50,000
\$5,000
r  10
1 12
McGraw-Hill/Irwin
\$50,000  \$5,000  (1  r )
12
 10
(1  r )  10
1 12
 1  1.2115  1  .2115
Calculator Keys

Texas Instruments BA-II Plus
FV = future value
 PV = present value
 I/Y = periodic interest rate

 P/Y
must equal 1 for the I/Y to be the periodic rate
 Interest is entered as a percent, not a decimal
N = number of periods
 Remember to clear the registers (CLR TVM) after
each problem
 Other calculators are similar in format

McGraw-Hill/Irwin
Multiple Cash Flows


Consider an investment that pays \$200 one
year from now, with cash flows increasing by
\$200 per year through year 4. If the interest
rate is 12%, what is the present value of this
stream of cash flows?
If the issuer offers this investment for \$1,500,
should you purchase it?
McGraw-Hill/Irwin
Multiple Cash Flows
0
1
200
2
3
4
400
600
800
178.57
318.88
427.07
508.41
1,432.93
McGraw-Hill/Irwin
Present Value < Cost → Do Not Purchase
Valuing “Lumpy” Cash Flows
First, set your calculator to 1 payment per year.
Then, use the cash flow menu:
CF0
0
CF3
600
I
CF1
200
F3
1
NPV
F1
1
CF4
800
CF2
400
F4
1
F2
1
McGraw-Hill/Irwin
12
1,432.93
4.3 Compounding Periods
Compounding an investment m times a year for
T years provides for future value of wealth:
r 

FV  C0  1  
m

McGraw-Hill/Irwin
mT
Compounding Periods
 For example, if you invest \$50 for 3 years at
12% compounded semi-annually, your
investment will grow to
.12 

FV  \$50  1 

2 

McGraw-Hill/Irwin
23
 \$50  (1.06)  \$70.93
6
Effective Annual Rates of Interest
A reasonable question to ask in the above
example is “what is the effective annual rate of
interest on that investment?”
FV  \$50  (1 
.12
)
23
 \$50  (1.06)  \$70.93
6
2
The Effective Annual Rate (EAR) of interest is
the annual rate that would give us the same
end-of-investment wealth after 3 years:
\$50  (1  EAR)  \$70.93
3
McGraw-Hill/Irwin
Effective Annual Rates of Interest
FV  \$50  (1  EAR)  \$70.93
3
(1  EAR) 
3
\$70.93
\$50
13
 \$70.93 
EAR  

 \$50 
 1  .1236
So, investing at 12.36% compounded annually
is the same as investing at 12% compounded
semi-annually.
McGraw-Hill/Irwin
Effective Annual Rates of Interest

Find the Effective Annual Rate (EAR) of an
18% APR loan that is compounded monthly.

What we have is a loan with a monthly
interest rate rate of 1½%.

This is equivalent to a loan with an annual
interest rate of 19.56%.
r 

1  
m

McGraw-Hill/Irwin
nm
12
.18 

 1 

12 

 (1.015)
12
 1.1956
EAR on a Financial Calculator
Texas Instruments BAII Plus
keys:
description:
[2nd] [ICONV]
Sets 12 payments per year
[↑] [C/Y=] 12 [ENTER]
Sets 18 APR.
[↓][NOM=] 18 [ENTER]
[↓] [EFF=] [CPT]
McGraw-Hill/Irwin
19.56
Continuous Compounding
The general formula for the future value of an
investment compounded continuously over many
periods can be written as:
FV = C0×erT
Where
C0 is cash flow at date 0,

r is the stated annual interest rate,
T is the number of years, and
e is a transcendental number approximately equal
to 2.718. ex is a key on your calculator.
McGraw-Hill/Irwin
4.4 Simplifications

Perpetuity


Growing perpetuity


A stream of cash flows that grows at a constant rate
forever
Annuity


A constant stream of cash flows that lasts forever
A stream of constant cash flows that lasts for a fixed
number of periods
Growing annuity

McGraw-Hill/Irwin
A stream of cash flows that grows at a constant rate for
a fixed number of periods
Perpetuity
A constant stream of cash flows that lasts forever
0
PV 
PV 
C
C
C
1
2
3
C
C
C
(1  r )

(1  r )
2

(1  r )
…
3

C
r
McGraw-Hill/Irwin
Perpetuity: Example
What is the value of a British consol that
promises to pay £15 every year for ever?
The interest rate is 10-percent.
0
£15
£15
£15
1
2
3
PV 
£15
…
 £150
.10
McGraw-Hill/Irwin
Growing Perpetuity
A growing stream of cash flows that lasts forever
0
PV 
PV 
McGraw-Hill/Irwin
C
C×(1+g)
C ×(1+g)2
1
2
3
C
(1  r )

C  (1  g )
(1  r )
2

C  (1  g )
(1  r )
3
…
2

C
rg
Growing Perpetuity: Example
The expected dividend next year is \$1.30, and
dividends are expected to grow at 5% forever.
If the discount rate is 10%, what is the value of this
promised dividend stream?
\$1.30
\$1.30×(1.05)
\$1.30 ×(1.05)2
…
0
1
PV 
McGraw-Hill/Irwin
2
\$1.30
.10  .05
3
 \$26.00
Annuity
A constant stream of cash flows with a fixed maturity
C
C
C
C

0
PV 
1
2
3
C
C
C
(1  r )

(1  r )
2

(1  r )
T
3

C
(1  r )
T

C
1
PV  1 
T 
r 
(1  r ) 
McGraw-Hill/Irwin
Annuity: Example
If you can afford a \$400 monthly car payment, how
much car can you afford if interest rates are 7% on 36month loans?
\$400
\$400
\$400
\$400

0
1
2
3
36

\$400 
1
PV 
 \$12,954.59
1 
36 
.07 / 12  (1  .07 12) 
McGraw-Hill/Irwin
What is the present value of a four-year annuity of \$100
per year that makes its first payment two years from today if the
discount rate is 9%?
4
PV1  
t 1
\$297.22
\$100
(1.09)
\$323.97
0
t

\$100
1
(1.09)
\$100
1
2
\$327 .97
 \$297 .22
PV 
McGraw-Hill/Irwin
0
1.09

\$100
(1.09)
2

\$100
3
\$100
(1.09)
3

\$100
(1.09)
\$100
4
4
 \$323.97
\$100
5
Growing Annuity
A growing stream of cash flows with a fixed maturity
C
C×(1+g) C ×(1+g)2
C×(1+g)T-1

0
PV 
1
C
(1  r )
2

3
C  (1  g )
(1  r )
2

T

 1 g  
C
 
PV 
1  
r  g   (1  r )  


McGraw-Hill/Irwin
T
C  (1  g )
(1  r )
T 1
T
Growing Annuity: Example
A defined-benefit retirement plan offers to pay \$20,000 per year
for 40 years and increase the annual payment by three-percent
each year. What is the present value at retirement if the discount
rate is 10 percent?
\$20,000
\$20,000×(1.03) \$20,000×(1.03)39

0
1
2
40
40

\$20,000
 1.03  
PV 
   \$265,121.57
1  
.10  .03   1.10  
McGraw-Hill/Irwin
Growing Annuity: Example
You are evaluating an income generating property. Net rent is
received at the end of each year. The first year's rent is
expected to be \$8,500, and rent is expected to increase 7%
each year. What is the present value of the estimated income
stream over the first 5 years if the discount rate is 12%?
\$8,500  (1.07) 
\$8,500  (1.07) 
3
\$8,500  (1.07) 
\$8,500  (1.07) 
2
\$8,500
0
1
\$34,706.26
McGraw-Hill/Irwin
\$9,095 \$9,731.65
2
3
4
\$10,412.87 \$11,141.77
4
5
4.5 What Is a Firm Worth?


Conceptually, a firm should be worth the
present value of the firm’s cash flows.
The tricky part is determining the size, timing
and risk of those cash flows.
McGraw-Hill/Irwin
Quick Quiz





How is the future value of a single cash flow
computed?
How is the present value of a series of cash flows
computed.
What is the Net Present Value of an investment?
What is an EAR, and how is it computed?
What is a perpetuity? An annuity?
McGraw-Hill/Irwin